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The Question:

The temperature $u(x,t)$ in a semi-infinite conductor occupying $x \in [0,\infty)$ satisfies the equation

$$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} \qquad x,t>0$$

The temperature is $0$ at $t=0$, i.e. $u(x,0)=0$.

For $t>0$, heat is supplied at $x=0$ at the constant flux

$$Q=-ku_x(0,t) \qquad k,Q \; \text{are constants}$$

By applying the Laplace Transform in the $t$ direction, find $u(x,t)$ at $x=0$ for $t>0$.


My Attempt:

I transformed the PDE, using the fact that $u(x,0)=0$:

$$p\hat u(x,p) = \frac{\partial ^2 \hat u}{\partial x^2}(x,p) \qquad p>0$$

and found the general solution:

$$\hat u(x,p) = A(p)e^{\sqrt p x}+B(p)e^{-\sqrt px}$$

Next, I transformed the boundary condition:

$$\frac{\partial \hat u}{\partial x}(0,p) = -\frac{Q}{k}p$$

BUT THE PROBLEM IS that there is only one boundary condition. Even if the question only asks for me to determine $u(x,t)$ at $x=0$, I still need one more boundary condition to determine both $A(p)$ and $B(p)$.

Am I misunderstanding something?

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  • 1
    $\begingroup$ Well, one part of the LT is exponentially growing. Is that going to work? $\endgroup$ – Ron Gordon May 31 '18 at 23:14
  • $\begingroup$ So basically, $A(p)=0$? $\endgroup$ – glowstonetrees May 31 '18 at 23:20
  • $\begingroup$ What should we expect the temperature to be like at any time as $x$ approaches infinity? $\endgroup$ – AEngineer May 31 '18 at 23:20
  • $\begingroup$ I think that would be the case. $\endgroup$ – Ron Gordon May 31 '18 at 23:21
  • $\begingroup$ $$\lim_{x \rightarrow 0}u(x,t)=0 \implies A(p) \equiv 0$$ $\endgroup$ – glowstonetrees May 31 '18 at 23:21

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